Method and apparatus for indicating rotational speed of footballs

ABSTRACT

Indicia on the surface of a football (for the American variety of the sport) are provided to provide visual means for assessing the spin speed of the football when thrown in a desired manner. In one embodiment, the indicia comprise a circumferential indicator ring consisting of a plurality of alternatingly-colored segments, the indicator ring being aligned with the desired direction of spin of the football. The segments are preferably selected to be of contrasting colors, and the width of the segments is selected such that the indicator ring appears to an observer to be a single, mixed color provided the football is spinning about its axis at at least a predetermined frequency. In another embodiment, the indicia comprise two or more circumferential indicator rings at least one of which comprising a plurality of alternatingly-colored segments. The respective segment widths on one or more rings may be selected such that the mixed color of one ring, and hence the spin speed of the football, can be assessed relative to the color or mixed color of another ring.

FIELD OF THE INVENTION

The present invention relates generally to footballs having a generally prolate spheroid shape, and more particularly relates to indicating the rotational speed of a thrown football by means of a visual indicator.

BACKGROUND OF THE INVENTION

In the American, Canadian, and Australian varieties of the game of football, as well as in the game of rugby, a ball having a generally prolate spheroid shape is used. (Australian footballs and rugby footballs tend to have more rounded ends than American and Canadian footballs. As used herein, the term “football” shall be understood to refer to any ball having a generally prolate spheroid shape, regardless of the particular sport for which it is intended). To produce a smooth and efficient trajectory, a football is preferably thrown such that it rotates or spins about its long axis (i.e., the axis through the ball's pointed ends). This spin is understood to generate forces that minimize wobbling of the ball, leading to a smooth, accurate, and energy-efficient trajectory. A professional football player can give a football a spin speed of about 10 revolutions per second. See, e.g., Wattsa et al., “The Drag Force on an American Football,” Am. J. Phys., Vol. 71, No. 8; August 2003.

In general, throwing a football accurately and efficiently is not an easily learned skill. A portion of the necessary skill involves achieving rapid spinning of the ball about its long axis. Both a beginner quarterback and a well skilled one are concerned with the spin speed of the ball. Consequently, for the purpose of developing and refining such skill, it would be advantageous to have a means of measuring the spin speed of a football during its flight.

Heretofore, there have not been shown any practical means of indicating or measuring the spin speed of a thrown football. More specifically, it is believed by the inventors that the prior art provides no practical methods for one to measure a football's spin speed or to evaluate progress one has made in achieving desired levels of spin on a football.

It is known in the prior art to provide multiple colors and other indicia on the surface of a football. However, it is believed that this is typically done primarily or exclusively for ornamental or cosmetic reasons, and not to provide information about a desired spin speed.

SUMMARY OF THE INVENTION

In view of the foregoing, the present invention is directed to a method and apparatus for indicating the rate of spin of a thrown football.

In accordance with one embodiment of the invention, indicia in the form of one or more indicator rings, comprising segments of alternating colors, are applied in some manner to the surface of a football. The indicia are functionally aligned with the desired spin direction of the ball. In one embodiment, the colors of an indicator ring and the widths of an indicator ring's alternatingly-colored segments are selected so that if the ball is spinning faster than a nominal speed, then the two colors of the ring appear to an observer to be one color, under nominal outdoor light conditions. In this disclosure, the term “mixed color” is used to refer to the observed single color created by the spinning of an indicator ring applied to a spinning object such as a football.

In accordance with one aspect of the invention, if an indicator ring applied to a football is observed to be substantially a single mixed color, this confirms to an observer that the spin speed of the football must be faster than a nominal speed.

In accordance with another aspect of the invention, a practicing quarterback can assess his/her progress in perfecting passing skills by assessing whether a football has been thrown with sufficient spin as to cause an indicator ring to appear to be one color instead of two colors.

In accordance with another aspect of the invention, a football having a two-color indicator ring as disclosed herein provides a means for both beginning and advanced quarterbacks to consistently evaluate spin speed and thereby assess his/her progress in developing passing skills.

In accordance with one aspect of the invention, the indicia on the surface of a football are provided to provide visual means for assessing the spin speed of the football when thrown in a desired manner. In one embodiment, the indicia comprise a circumferential indicator ring consisting of a plurality of alternatingly-colored segments, the indicator ring being aligned with the desired direction of spin of the football. The segments are preferably selected to be of contrasting colors, and the width of the segments is selected such that the indicator ring appears to an observer to be a single, mixed color provided the football is spinning about its axis at at least a predetermined frequency. In another embodiment, the indicia comprise two or more circumferential indicator rings at least one of which comprising a plurality of alternatingly-colored segments. The respective segment widths on one or more rings may be selected such that the mixed color of one ring, and hence the spin speed of the football, can be assessed relative to the color or mixed color of another ring.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is best understood with reference to the following detailed description of various embodiments of the invention, when read in conjunction with the accompanying drawings, in which like numerals refer to like elements, and in which:

FIG. 1 is a side view of a football having spin speed indicia in accordance with one embodiment of the invention applied thereon;

FIG. 2 is an end view of the football from FIG. 1;

FIG. 3 is a side view of a football having spin speed indicia in accordance with another embodiment of the invention applied thereon;

FIG. 4 is an end view of the football from FIG. 3;

FIG. 5 is a side view of a football having spin speed indicia in accordance with another embodiment of the invention applied thereon;

FIG. 6 is an end view of the football from FIG. 5;

FIG. 7 is a side view of a football having spin speed indicia in accordance with another embodiment of the invention applied thereon;

FIG. 8 is an end view of the football from FIG. 7;

FIG. 9 is a perspective view of a Newton's Surface in the form of a color wheel;

FIG. 10 is a perspective view of a Newton's Surface in the form of a cylinder;

FIG. 11 is a perspective view of a Newton's Surface in the form of an American football;

FIG. 12 is a perspective view of a football having circumferential indicia applied thereon;

FIG. 13 is a perspective view of a cylinder illustrating angular and spatial relations of certain features thereof;

FIG. 14 is a perspective view of a football illustrating angular and spatial relations of certain features thereof;

FIG. 15 a is an illustration of spin speed indicia for a football, in accordance with one embodiment of the invention;

FIG. 15 b is an illustration of spin speed indicia for a football, in accordance with an alternative embodiment of the invention;

FIG. 15 c is an illustration of spin speed indicia for a football, in accordance with an alternative embodiment of the invention;

FIG. 15 d is an illustration of spin speed indicia for a football, in accordance with an alternative embodiment of the invention;

FIG. 15 e is an illustration of spin speed indicia for a football, in accordance with an alternative embodiment of the invention; and

FIG. 16 is an illustration of spin speed indicia for a football, in accordance with an alternative embodiment of the invention.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS OF THE INVENTION

In the disclosure that follows, in the interest of clarity, not all features of actual implementations are described. It will of course be appreciated that in the development of any such actual implementation, as in any such project, numerous engineering and technical decisions must be made to achieve the developers' specific goals and subgoals (e.g., compliance with system, technical, and practical constraints), which will vary from one implementation to another. Moreover, attention will necessarily be paid to proper design and engineering practices for the environment in question. It will be appreciated that such development efforts could be complex and time-consuming, outside the knowledge base of typical laymen, but would nevertheless be a routine undertaking for those of ordinary skill in the relevant fields.

Referring to FIGS. 1 and 2 a first embodiment of the current invention is described. In FIGS. 1 and 2, a football 1 is depicted with respect to a longitudinal axis pp′. Football 1 has two ends 201 and 202, and pp′ axis passes through the ends 201 and 202.

In accordance with one embodiment of the invention, football 1 has spin speed indicia in the form of a circumferential indicator ring 2 applied on its surface. In the presently disclosed embodiment, indicator ring 2 comprises alternatingly-colored segments of a first color 3 (3 a-3 e) and a second color 4 (4 a-4 e) substantially aligned with the desired spin direction of football 1 about its long axis. Each segment 3 is placed between two segments 4, and each segment 4 is placed between two segments 3.

In accordance with one embodiment, segments 3 and segments 4 have the same width, W. Preferably, width W is chosen so that there are an equal number of segments 3 and segments 4. In one exemplary embodiment, segments 3 are yellow and segments 4 are blue. In one embodiment, five segments of each color are provided (i.e., N=5). FIG. 2 depicts football 1 in the direction of pp′ axis. It is to be noted that in FIGS. 1 and 2, W is measured along the surface of the ball.

When football 1 is thrown with a spin speed of R (measured in rotations per second or RPS) about axis pp′, an observer sees segments 3 and segments 4 moving consecutively with frequency F, measured in Hertz, where

F=R*N (Hertz, Hz)

The time it takes for one composite segment consisting of one segment 3 and one segment 4 to replace the next such composite segment in the progression is 1/F seconds.

If frequency F is large enough, then segments 3 and segments 4 together appear to an observer to be a single, mixed color. As would be known to persons having ordinary skill in the art, the resulting color is referred to as the mixed color of the two respective original colors.

In a system producing a repetitive, periodic pattern of colors, the smallest frequency that generates a mixed color is called the mixing frequency, Fm, of the system. In the prior art, the mixing frequency is sometimes called the critical flicker frequency. Herein, the term mixing spin speed Rm (RPS) of a ring is used to denote the rotational spin speed R corresponding to the mixing frequency Fm. Therefore, the mixing spin speed Rm and the mixing frequency Fm are related as follows:

Fm=Rm*N (Hz)

where N is the equal number of segments of each color comprising the indicator ring (for example, N=5).

As would be appreciated by those of ordinary skill in the art, the concept of a “mixing frequency” implicates the more general concept of a so-called Newton's surface. A Newton's surface traditionally refers to any surface having a rotational axis and having a pattern of alternating first and second colors applied thereon. When the surface is rotated about its axis, the colors on a Newton's surface appear blended or mixed when the surface is rotating at at least a certain speed; the higher the rotational speed the more complete the blending of the colors. Nevertheless, for any given average mean brightness, there is a rotational speed, ms, measured in revolutions per second (RPS), above which no more blending occurs. This speed is called the mixing spin speed (RPS), ms, of the colored pattern on the surface.

At the mixing spin speed, all points in a plane perpendicular to the rotational axis at a distance x from the origin of the rotational axis, and at a distance, d, from the rotating axis appear to be the same, single color. This color is called the mixed color, mc, of the colored surface at distance (x, d).

More specifically, a Newton's wheel, a Newton's cylinder, and a Newton's Ball are referred to Newton's surfaces when the surface is a circle or a wheel, a cylinder, and a football, respectively. FIGS. 9-11 depict a Newton's wheel 35, a Newton's cylinder 31, and a Newton's ball 33, respectively. Each surface has an axis of rotation pp′. The origin of the axis pp′ is denoted by O.

At a distance (x,d), the mixing colors of the three Newton's surfaces: Wheel, Cylinder, and Ball are denoted by mc1, mc2, and mc3, respectively.

Referring to FIG. 10 a Newton's cylinder 31 is shown comprising end points 216 and 217, and an indicator ring 49. FIG. 10 also depicts axis of rotation pp′ having origin O, and a point 218 at distance (x, d), where x is measured from O to the plane perpendicular to pp′ and passing through the point 218, and d is measured from point 218 to the point of intersection of pp′ and the said perpendicular plane. Ring 49 has color segments 41 and 42. The colored segments 41 and 42 have width W. Finally the perpendicular plane splits ring 49 into two identical rings.

Referring to FIG. 11 a Newton's ball 33 is shown comprising end points 213 and 214, and a ring 50. FIG. 11 also depicts axis of rotation pp′ having origin O, and a point 215 at distance (x, d), where x is measured from O to the plane perpendicular to pp′ and passing through the point 215, and d is measured from point 215 to the point of intersection of pp′ and the said perpendicular plane. Ring 50 has color segments 43 and 44.

The width of the ring 50, referred to as WR, is defined as follows: consider two planes perpendicular to pp′—the first plane touching ring 50 on its side closer to the end 213, and the second plane touching the ring 50 on its side farther away from the end 213. Now the distance between these two planes is defined to be the width WR of ring 50. Note that WR is not related to W, width of the color segments.

Consider further a third plane perpendicular to pp′ and between the first two planes and equidistant to both planes. The intersection of the third plane with ring 50 is shown using a is dashed circle in FIG. 11. This circle is called “the center line” of the ring 50.

For a Newton's ball the width, W, of the colored segments 43 and 44 is defined as for FIG. 11. Note that in this case, W is measured along the dashed circle.

Referring to FIG. 9, a Newton's circle or wheel 35 is shown comprising pizza slice shape, color segments 32 and 40. FIG. 9 also depicts axis of rotation pp′ having origin O, and a point 221 at distance (x, d), where x is measured from O to the plane perpendicular to pp′ and passing through the point 221. Since the perpendicular plane to pp′ that passes the point 221 also passed O, then x=0. And d is measured from point 221 to the point of intersection of pp′ and the perpendicular plane.

Consider Newton's ball 33 of FIG. 11. Ball 33 has ring 50 comprising alternating segments 43 having color cl and segments 44 having color c2. The number of c1 (or c2) segments is denoted by N, (N=4).

Ring 50 is placed on the ball such that the points on its center line are at distance (x, d). Another way of saying this is that a plane perpendicular to pp′ at the distance x, with respect to the origin O, would cut ring 50 through the center line of ring 50.

It is to be noted that:

-   -   A. For Newton's ball 33, parameter d is a function of x. When x         is such that the perpendicular plane is closer to ends 213 or         214 of the football, then d is small, but when x is such that         the plane is closer to the middle of the football, then d is         large.     -   B. Referring to FIG. 11, define the circumference of ring 50 to         denote the length of its center line. Therefore, circumference         of ring 50=2*N*W, where W is measured along the center line. Now         if W and N are fixed, then the circumference of the ring is         fixed and therefore, it can only be placed fully on a location         on the football having the same circumference. Hence, either         zero, one or two places on the football can accommodate the ring         50. More specifically, if the circumference of ring 50 is less         than the circumference of the ball at its middle, then ring 50         can be placed on either side of the middle of ball 33 at a         unique location having the same circumference. But if the         circumference of ring 50 is greater than the circumference of         the ball at its middle, then ring 50 can't be placed on ball 33.         Finally if the circumference of ring 50 is exactly equal to the         circumference of the ball at its middle, then ring 50 can be         placed at the middle of ball 33 and at no other location.     -   C. On the other hand, if the size of the football is fixed, and         assuming a distance (x,d), then the parameter W, width of the         segments, becomes a function of N. For a fixed distance (x,d),         Wi is denoted by the width of color segments of a ring similar         to ring 50, when N=i, i=1,2, and so on.     -   D. Therefore, referring to FIG. 12, for two rings 101 and 102 on         ball 34, where ring 101 has parameter N1 and it is at distance         (x1,d1) while ring 102 has parameter N2 and it is at distance         (x2,d2), if d1<d2 as shown in FIG. 12, then if N1=N2, then         W1=W(ring 101)<W(ring 102)=W2.

If Newton's ball 33 of FIG. 11 is rotated faster and faster about its axis pp′, after the rotation spin speed exceeds a threshold, ms RPS, the colors of the ring 50 will appear to be a single mixed color, mc. The parameters, ms and mc, are the mixing spin speed and the mixed color of the ball 33, respectively, related to the colors on ring 50.

Ring 50 of FIG. 11 has characterization:

[{c1 c2}, c, s}], if c=mc, and s=ms.

In addition, for every spin speed, s, define a parameter f, f=s*N, where N is as before. The parameter f is measured in Hertz, Hz, and it refers to the number of times a pattern is repeated in one second with respect to a stationary point close to the surface of a Newton's surface.

For ring 50 of FIG. 11, and for a given average brightness, one can obtain mc and ms with at least two methods:

Method 1) Use a Newton's ball. Increase the spin speed until the colors of ring 50 appear fully mixed. The mixing spin speed=ms=the minimum speed when the colors of ring 50 appear fully mixed, and the mixing color=mc=the color of ring 50 when its colors appear fully mixed.

Method 2) Human eye perception has been studied extensively. For many average brightness levels and colors, eye response is measured with respect to (a) frequency, f; (b) spatial frequency fs; or (c) both. In general, for a stationary picture, the spatial frequency, fs, is the number of times a repeated pattern falls inside one degree of the viewing angle. More specifically, referring to FIG. 11, assume the observer is at a distance D from the Newton's ball looking at the ball in a direction consistent with FIG. 11 illustration. Also assume ball 33 is stationary, i.e., it is not turning, then fs=the number segments of the same color that the observer sees in one degree angle.

The measured eye responses give the so called flicker frequencies, fk, and fks. With respect to frequency, f, color mixing occurs at the flicker frequency, fk. And with respect to fs, when the ring is not rotating, the repeated patterns appear uniform and mixed at the flicker frequency, fks.

Those having ordinary skill in the art will appreciate that there is a dependency between flicker frequencies, fk and fks. In general, the flicker frequency, fk, drops when spatial frequency is increased.

One way to increase spatial frequency is to place more segments on a given ring, i.e. smaller W, or equivalently larger N. Another way to increase spatial frequency is to view the same ring from a farther distance—larger D. Yet another way to increase, fs, for a given ring is to keep N fixed but reduce W, then to place the ring closer to one of the ends of the ball.

Clearly, for a given averaged mean brightness and a ring, a Newton's surface can be used to characterize the dependency between the flicker frequency, fk, (and therefore, ms) and the spatial frequency. One can place the surface at different distances and measure the mixing spin speed (or the mixing frequency).

Fortunately, and in accordance with one aspect of the invention, for ms belonging to a wide range of values, and for mean brightness values corresponding to many outdoor, day conditions, the flicker frequency, fk, of a ring for a football does not vary significantly for observers at a few yards to a few tens of yards from the ring. This is easily verified with the help a Newton's surface.

It is to be noted that:

-   -   (1) in most indoor or dark outdoor conditions when average mean         brightness is low, the flicker frequency, fk, is more sensitive         to the spatial frequency, fs, variations.     -   (2) the flicker frequency, fk, is more likely to be         significantly affected by the spatial frequency, fs, when for a         fixed N, a ring is placed too close to one of the ends of the         ball since as discussed earlier, W becomes small.

Now, if the flicker frequency, fk, is known, then we can obtain the mixing spin speed from the equation below.

ms=fk/N.

In method (2) above it was shown that:

Given an average brightness and an N, first flicker frequency, fk, can be derived. Then ms can be derived using ms=fk/N. The parameter mc can be obtained using a Newton;s surface.

The following shows how to obtain N for ring 50 of FIG. 11 given the following information:

-   a) average brightness, -   b) flicker frequency, fk, and -   c) a desired mixing spin speed, ms.     To obtain N, use

N=fk/ms.

If N is not close to an integer, one of the following solutions can be applied.

Solution (1): Approximate N by ceiling(N), where ceiling(N)=the smallest integer larger than or equal to N (Note that N=fk/ms). This selection for N would insure that when the spin speed reaches ms, the frequency, f, f=N*s, (s=ms) is larger than the flicker frequency. This implies that the colors are appearing mixing. Solution (2): Approximate N by floor(N), where floor(N)=the largest integer smaller than or equal to N (Again note that N=fk/ms). Divide the circumference of the ring by N=fk/ms.

Let b=(circumference of the ring)/(fk/ms).

Choose floor(N) segments with color c1 and width W=b, and choose floor(N) segments with color c2 and width W=b, then place them on the ring alternatively.

Cover the remaining portion of the ring by color mc. Instead of coloring the remaining portion mc, it can be covered with shorter width color segments having mixed color equal to mc and having mixed speed slower than ms. To the same effect, the remaining portion of the ring can be covered as described below with reference to FIGS. 5 and 6. Solution (2) insures that the color mixing happen at spin speeds close to ms but it forgoes the simple periodic color pattern of the ring.

Solution (3): Choose colors, c1 and c2, which have a different flicker frequency, fk, for the given average brightness. More specifically, choose colors c1 and c2 whose flicker frequency produces an N which is close to an integer.

Any ring of more than one color situated at a distance (x, d) on a Newton's surface will produce a mixed color at some spin speed. The mixing spin speed, ms, and the mixing color, mc, can be found with the help of the Newton's surface.

For example, the colored segments on a ring can have segment shape as before or they can have more complex shapes as suggested below.

For the sake of the present disclosure, first consider segments for a ring on a Newton's cylinder, and then map the segments to a ring on a Newton's ball. To this end, two functions, function 1 and function 2, are defined that can map segments from a Newton's cylinder to a Newton's ball.

Function 1: Referring to FIG. 13, a cylinder 55 is illustrated with respect to an xyz coordinate system. Cylinder 55 intersects the x-axis at (a, 0, 0) coordinate. Cylinder 55 intersects the y-axis at (0, a, 0) coordinate.

Cylinder 55 is characterized by the following equation.

(x ² +y ²)/a ²=1

Referring to FIG. 14, a ball 56 is illustrated with respect to an xyz coordinate system. Ball 56 intersects the x-axis at (a, 0, 0) coordinate. Ball 56 intersects the y-axis at (0, a, 0) coordinate. Ball 56 intersects the z-axis at (0, 0, b) coordinate.

Cylinder 55 is characterized by the following equation.

(x ² +y ²)/a ² +z ² /b ²=1

Given a point A on cylinder 55, the following parameters are defined:

Az=distance of point A to xy plane

t=the angle between x-axis and the line passing through the origin, R, of the xyz coordinates, and the point A′, which is the projection of A onto xy plane. The angle is measured counterclockwise in the xy plane looking from positive z half of the space.

Function 1 maps point A to a point B on ball 56, having coordinates (Bx, By, Bz) where

Bx=sqrt(a ²×(1−Az ² /b ²))×cos(t)

By=sqrt(a ²×(1−Az ² /b ²))×sin(t)

Bz=Az

Function 1 maps a segment, Q1, on cylinder 55 to a segment, Q2, on ball 56 by mapping the points on Q1 to points on ball 56. Of course in general not every point need to be mapped and only enough points to enable a close approximation of Q2.

Function 2: To describe function 2, use FIGS. 13 and 14 as before except do not assume the following characterization of ball 56.

(x ² +y ²)/a ² +z ² /b ²=1

Given a point A on cylinder 55, we define parameters Az and t as before.

Function 2 maps point A to a point B on ball 56, where B has coordinates (Bx, By, Bz) and

Bx=cos(t)×Circumference(Az)/(2×pi)

By=sin(t)×Circumference(Az)/(2×pi)

Bz=Az

where Circumference(Az) is the circumference of ball 56 measured at z=Az.

Function 2 maps any segment, Q1, on cylinder 55 to a segment, Q2, on ball 56.

Below color segments are described for rings on Newton's cylinder. Each segment generates a segment on Newton's ball using function 1 or function 2.

Briefly,

-   (1) Color segments can be segments or closed segments. -   (2) Segments can be identical or not.     21 More specifically, color segments can belong to any class or     subclass below:

A. Identical segments

-   -   a. of rectangular shape, as shown in FIG. 15 a.     -   b. of parallelogram shape, as shown in FIG. 15 b.     -   c. with curved boundaries, as shown in FIG. 15 c.

B. Non-identical segments

-   -   a. shown in FIG. 15 d as having two colors: c1 and c2.

C. Identical closed segments

-   -   a. of triangular shape, as shown in FIG. 15 e.

Referring to FIG. 15 e, a band, 300, comprising triangles of two colors is shown. If band 300 is cut into narrower bands 301, 302 and 303, then bands 301 and 303 would have higher mixed spin speeds than ring 302. Ring 300 will appear having uniform color when all its narrow bands reach their mixed spin speeds. In another words, when spin speed reaches the mixed spin speed of a sub-band of ring 300 having the largest spin speed, then ring 300 will appear having mixed colors. However, since different bands might have different mixed spin speeds, the sub-bands with lower mixed spin speed will appear mixed before the sub-bands with larger mixed spin speeds. These variations provide an advantage in that the lower mixed spin speed bands would provide “a reference color” for the larger mixed spin speed bands—similar to ring 5 and ring 6 (described below), respectively.

D. More generally any pattern that produces a mixed color at a predetermined spin speed can be depicted on the ring.

In one embodiment, the two colors of an indicator ring 2 from FIG. 1 are yellow and blue, and the mixed color is green. The mixing frequency ranges from 20 to 80 Hz, depending on the mean brightness conditions—the brighter the conditions, the higher the critical frequency. See, Yao Wang et al., “Video Processing and Communications,” ISBN No. ______, Wiley, 2005. If average mean brightness requiring a critical flicker frequency of 50 Hz is assumed, then a spin speed,

R=Fm/N=50 Hz/5=10 (RPS)

will produce the mixed color, green, to the observer. Therefore, Rm=10 RPS under the brightness assumption above.

In accordance with one aspect of the invention, football 1 helps players practice achieving a desired spin speed of a thrown ball. If segments 3 and segments 4 do not produce the appearance of the mixed color, then players would know that additional spin speed is required. On the other hand, if the mixed color is produced, then players are provided positive feedback regarding their spin speed skill.

In addition, players working on their spin speed can measure their progress when they produce the mixed color if initially failing to do so.

The following notation summarizes some of the color mixing parameters of football 1.

A ring similar to ring 2 in FIGS. 1 and 2 has characteristic I given by

I(ring)=[{c1,c2}, m, s]

if the two colors are c1 and c2, the mixed color is m, and the mixing spin speed is s (RPS), for a given brightness.

Using this convention, an indicator ring 2 in an exemplary embodiment may have the following characteristic:

I(ring 2)=[{yellow, blue}, green, 10 RPS]

assuming an average mean brightness requiring a critical flicker frequency of 50 Hz.

Those of ordinary skill in the art having the benefit of the present disclosure will appreciate that in order to produce a slower mixing spin speed for a given football 1, the characteristic N can be increased. If N is increased from five to ten, for example, then the mixing spin speed Rm from the above example becomes

Rm=Fm/N=50/10=5 RPS.

This is one-half of the initial mixing spin speed (10 RPS). Lower mixing spin speeds might often be desirable. For example, a quarterback might plan to acquire the skill of spinning 10 RPS by practicing with five footballs 1 a, 1 b, 1 c, 1 d, and 1 e having mixing spin speeds 6 RPS, 7 RPS, 8 RPS, 9 RPS, and 10 RPS, respectively. He would start with football 1 a and then switch to football 1 b when he feels comfortable spinning 6 RPS. Then he would switch to football 1 c and so on as he masters each ball.

Also for a child, football player a goal of achieving a small mixing spin speed obviously is more practical than a large mixing spin speed.

Turning now to FIGS. 3 and 4, there is shown a football 10 in accordance with an alternative embodiment of the invention. In particular, FIG. 3 illustrates a football 10 having ends 203 and 204, and having an indicator ring 2 substantially the same as in the embodiment of FIGS. 1 and 2 (and hence retaining the same reference numeral in FIGS. 3 and 4). In addition to its ring 2, football 10 in FIGS. 3 and 4 has a second indicator ring 5. Ring 5 is preferably of a single color, and more specifically it is preferably the same color as the mixed color of ring 2.

Thus, for football 10, the characteristic I is given by:

I(ring 2)=[{yellow, blue}, green, 10 RPS]

where Color(ring 5)=green=the mixed color of ring 2. (Herein, the notation Color(X) is used to to denote the color of entity X.)

As previously described, assuming an average mean brightness, ring 2 of football 10 will appear to an observer to have its mixed color if football 10 has a spin speed of R=10 RPS. Therefore, in the embodiment of FIGS. 3 and 4, rings 2 and 5 would appear having the same color when R=10 RPS. Thus ring 5 provides a “reference color for the mixed color of ring 2 since Color(ring 5)=the mixed color m of ring 2.

Similar to football 1, football 10 helps players practice achieving a desired spin speed. However, for football 10 it is believed to be easier to recognize the mixing of colors of ring 2 having it juxtaposed with ring 5, which has the mixed color of ring 2.

Those of ordinary skill will appreciate that increasing of the spin speed beyond the mixing spin speed would not change the appearance of ring 2. That is, once colors appear mixed, any faster spin speed does not change that appearance.

Turning now to FIGS. 5 and 6, there is shown a football 20 in accordance with another 11 embodiment of the invention. FIG. 5 illustrates a football 20 having ends 205 and 206, and having ring 2 of the first embodiment. In addition to its ring 2, football 20 has a second ring 6. Ring 6 is similar to ring 2 except that the width, w, of its yellow and blue segments are smaller than W, the width of segments 3 and 4 of ring 2.

Therefore, football 20 has the following characteristics

I(ring 2)=[{yellow, blue}, green, 10 RPS], and

I(ring 6)=[{yellow, blue}, green, r RPS],

where r is much smaller than 10 RPS. For ring 6 of FIG. 5, r=10/3 RPS.

Again assuming an average mean brightness, ring 2 will appear to an observer as the mixed color if football 20 is spun with spin speed R=10 RPS. However ring 6 will appear as the mixed color at lower spin speeds, even at speed spin as small as r=10/3 revolution per second. Therefore, for spin speeds of interest—10 RPS for football 20—ring 6 will appear green and it can provide the reference color for the mixed color of ring 2.

Hence ring 6 in FIGS. 5 and 6 and ring 5 in FIGS. 3 and 4, respectively, perform the same function; namely that of they providing a reference color for the mixed color of ring 2. Note that r can easily be made smaller than 10/3 by reducing w.

An advantage of ring 6 in FIGS. 5 and 6 over ring 5 in FIGS. 3 and 4 is that ring 6 provides a more robust reference for the mixed color. That is, in FIGS. 3 and 4, if the mixed color (e.g. green) used in ring 5 differs from the “true” mixed color due to wear and tear or other reasons, then the observer will not see closely matched colors in rings 2 and 5.

FIGS. 7 and 8 illustrate still another embodiment of the present invention. FIG. 7 depicts a football 30 having a ring 2 substantially the same as ring 2 in the above-described embodiments. In addition to ring 2, football 30 has a second ring 9. Ring 9 comprises segments: 7 a-7 d, having a third color, c3, and segments: 8 a-8 d having a fourth color, c4. Each segment 7 is placed between two segments 8, and each segment 8 is placed between two segments 7. Segments 7 and 8 both have width W2, as shown in FIGS. 7 and 8. Again W2 is measured along the surface of football 30. Football 30 has ends 207 and 208.

Football 30 thus has the following characteristics:

I(ring 2)=[{yellow, blue}, green, 10 RPS], and

I(ring 9)=[{c3, c4}, m2, s2 RPS].

Ring 9 in the embodiment of FIGS. 7 and 8 differs from ring 5 in FIGS. 3 and 4 and ring 6 of FIGS. 5 and 6. In particular, ring 5 and ring 6 provide “a reference color” for the mixed color of ring 2. But ring 9 in the forth embodiment provides advantages explained below.

Again assuming an average mean brightness, ring 2 will appear to an observer as a mixed color if football 30 is spun with spin speed R=10 RPS.

There are eight variations to ring 9. These variations and their advantages are given in Table 1, where c1=yellow, c2=blue, m1=green, and s1=10 RPS

TABLE 1 Variation Parameters of ring 9 Advantage provided by ring 9 1 c3 = c1 more confirmation c4 = c2 m2 = m1 s2 = s1 2 c3 = c1 estimate progress c4 = c2 m2 = m1 s2 ≠ s1 3 c3 = c1 more confirmation c4 = c2 m2 ≠ m1 s2 = s1 4 c3 = c1 estimate progress c4 = c2 m2 ≠ m1 s2 ≠ s1 5 c3 ≠ c1 more confirmation c4 ≠ c2 m2 = m1 s2 = s1 6 c3 ≠ c1 estimate progress c4 ≠ c2 m2 = m1 s2 ≠ s1 7 c3 ≠ c1 more confirmation c4 ≠ c2 m2 ≠ m1 s2 = s1 8 c3 ≠ c1 estimate progress c4 ≠ c2 m2 ≠ m1 s2 ≠ s1

Variations 1-8 are explained below. Although N=5 for ring 2, and N=4 for ring 9, these numbers are chosen for illustration purposes and they are not assumed to be fixed.

Variation 1:

-   c3=c1; c4=c2; m2=m1; s2=s1; -   The two rings make it easier to observe the mixed color hence they     make it easier to validate achieving the mixing speed. This     variation can be realized with

N(ring 9)=N(ring 2).

Variation 2:

-   c3=c1; c4=c2; m2=m1; s2≠s1, s2>s1; -   The two rings have different mixing spin speeds; ring 2 reaches its     mixing spin speed sooner than ring 9. This variation provides a     method to evaluate one's progress in spinning football 30. If ring 2     appears color m1 but ring 9 colors, c3 and c4, do not appear mixed,     then this indicates to one the need to improve the spin. Progress is     confirmed when ring 9 appears color m2. Note in this variation     m2=m1. This variation can be realized with

N(ring 9)<N(ring 2).

Variation 3:

-   c3=c1; c4=c2; m2≠m1; s2=s1; -   The two rings have different mixed colors but the same mixing spin     speed. This variation is similar to variation 1; the two rings make     it easier to confirm that the mixing speed is achieved. This     variation can be realized when FOR THE same ring (ring 2 or ring 9),

W(one color segment)≠W(the other color segment).

Variation 4:

-   c3=c1; c4=c2; m2≠m1; s2≠s1, s2>s1; -   The two rings have different mixing spin speeds and different mixed     colors; ring 2 reaches its mixing speed sooner than ring 9. This     variation works similar to variation 2. If ring 2 appears m1 but     ring 9 colors, c3 and c4, do not appear mixed, then this indicates     the need to improve the spin. Progress is confirmed when ring 9     appears as color m2. Note in this variation that m2≠m1. This     variation can be realized when for the same ring, W(c3)≠W(c4).

Variation 5:

-   c3≠c1; c4≠c2; m2=m1; s2=s1; -   The two rings have different colors but the same mixed color and the     same mixing speed. The advantage of variation 5 is very similar to     that of variations 1 and 3. The two rings make it easier to observe     the mixed color hence they make it easier to validate achieving the     mixing speed.

Variation 6:

-   c3≠c1; c4≠c2; m2=m1; s2≠s1, s2>s1; -   The two rings have different colors and different mixing speeds;     again, ring 2 reaches its mixing speed sooner than ring 9. This     variation provides another method to evaluate one's progress in     spinning the football 30. If ring 2 appears color ml but for ring 9     colors, c3 and c4, do not appear mixed, then this indicates the need     to improve the spin. Progress is confirmed when ring 9 appears color     m2. Note in this variation m2=m1. This variation can be realized     with N(ring9)<N(ring2).

Variation 7:

-   (c1 c2)≠(c3 c4); m2≠m1; s2=s1; -   The two rings 2 and 9 do not have the same colors. That is, at least     one of their colors differ. Also they have different mixed colors     but the same mixing spin speed. This variation is similar to     variations 1, 3 and 5; the two rings make it easier to confirm that     the mixing speed is achieved.

Variation 8:

-   (c1 c2)≠(c3 c4); m2≠m1; s2≠s1, s2>s1; -   The two rings 2 and 9 do not have the same colors. That is, at least     one of their colors differ. Also 8 they have different mixing spin     speeds and different mixed colors; ring 2 reaches its mixing speed     sooner than ring 9. This variation works similar to variations 2, 4     and 6. If ring 2 appears m1 but ring 9 colors, c3 and c4, do not     appear mixed, then this indicates the need to improve the spin.     Progress is confirmed when ring 9 appears color m2. Note in this     variation m2≠m1. This variation can be realized with     N(ring9)<N(ring2).

In the description of the embodiments, width W was measured along one side of the ring. Those of ordinary skill having the benefit of the present disclosure will recognize that W may be measured according to various measuring conventions. Nonetheless, given a ring, its width, WR, and its location on the ball, as long as it is stated where W is measured there will be no ambiguity.

Although the embodiments described herein involve footballs with one or two rings, other variations or generalizations have been conceived, including: (I) a football with more than two rings; (II) a football with more than two rings where at least three of the rings have distinct mixed colors; (III) a football with more than two rings where at least three of rings: ring 1, ring 2, and ring 3 have distinct mixing spin speeds s1, s2, and s3, where s1<s2<s3. If ring 1 colors appear mixed but ring 2 and ring 3 colors do not appear mixed, then this indicates that spin speed s1 is achieved bt we need to practice to achieve s2 and s3. If ring 1 and ring 2 colors appear mixed but ring 3 colors do not, then this indicates further progress but with a need to practice to achieve s3. When all rings appear in their mixed color then the goal is achieved. Therefore, the combination of ring 1, ring 2, and ring 3 enable a more graduated method to measure progress than two rings.

It is also contemplated that football 30 of the embodiment of FIG. 7 may support an additional “reference ring” similar to ring 5 of the embodiment of FIG. 3. The added ring can provide “a reference color” corresponding to the mixed color of one of the rings: ring 2 or ring 9 of the embodiment of FIG. 7.

Furthermore, football 30 of the embodiment of FIG. 7 may support an additional “reference ring” similar to ring 6 of the embodiment of FIG. 5. The added ring can provide “a reference color” corresponding to the mixed color of one of the rings: ring 2 or ring 9 of the embodiment of FIG. 7.

In one embodiment of the invention, it is contemplated that a set, S, containing a predetermined spin speeds is generated. For example S={s1=2 RPS, s2=3 RPS, s3=4 RPS, s4=5 RPS}. The spin speeds in S are selected to be suitable for spin training in throwing a football. Next for each spin speed in the set S, a football according to one of the foregoing embodiments is produced. A trainee would start with the football with the ring having a mixing spin speed, s1. He would practice until he achieves spins causing the mixed color on the ring on the football. The trainee will then move up to the football with the ring having the mixing spin speed, s2, and so on.

In all given embodiments one may use a ring having three or more color segments juxtaposed in a sequential pattern. In terms of the nomenclature used herein, I(ring of three colors)=[{c1,c2,c3},m,s]: For example, a ring can have three colors c1, c2, and c3 where the colors are arranged periodically where c2 follows c1 and c3 follows c2 and then c1 follows c3 again.

As previously noted, the colored segments do not have to be segments. FIG. 16 depicts a ring 110 having colored segments comprising closed segments that are portions of six sided polygons. Ring 110 has the following representation:

I(ring 110)=[{c1,c2,c3}, m, s].

For a given average mean brightness, an easy way to find the mixing spin speed, ms, and 9 the mixed color, mc, of a ring with more than two colors from a distance D, (D=distance between the observer and the ring) is to use a Newton's surface as before.

Referring to FIG. 16, if one cuts a band 110 into narrower bands 111, 112, and 113, then bands 111 and 113 would have higher mixed spin speeds than band 112. Ring 110 will appear having uniform color when all its narrow bands reach their mixed spin speeds. In another words, when spin speed reaches the mixed spin speed of a sub-band of the ring 110 having the largest spin speed, then ring 110 will appear having mixed colors. However, since different bands might have different mixed spin speeds, the sub-bands with lower mixed spin speed will appear mixed before the sub-bands with larger mixed spin speeds. Again the variations on the mixed spin speed provide an advantage that the lower mixed spin speed bands would provide “a reference color” for the larger mixed spin speed bands, similar to ring 5 and ring 6 in the embodiments of FIGS. 3 and 5, respectively.

In another alternative embodiment, colored rings can be formed on footballs by painting the surface of the ball. They can also be formed by adhering a colored band on the football.

Also, if the band is elastic with enough gripping, then it can be fixed to the surface without using any adhesives. Elastic bands provide an additional advantage in that they can be removed and be replaced by other colored bands.

If the flicker frequency change is pronounced, due for example to a spatial frequency change resulting from the ball moving away from the observer, then the ball can be designed according to the embodiment of FIG. 7 (cases 2, 4, 6, or 8). More specifically, partition the special frequency into two sets: low frequency and high frequency. Select a flicker frequency, fk1, produced when spatial frequency is in low frequency set, and select a flicker frequency, fk2, produced when spatial frequency is in high frequency set. Now match ring 2 mixing spin speed to fk1, and match ring 9 mixing spin speed to fk2. All we need to do now is to observe ring 2 for color mixing when the ball is near and to observe ring 9 when the ball is far.

When applying the use of the present invention to a group of users having a same age, flicker frequency variations may be assumed negligible.

From the foregoing disclosure, it should be apparent that a method and apparatus for indicating spin speed of a thrown football has been disclosed.

Although a specific embodiment of the invention as well as possible variants and alternatives thereof have been described and/or suggested herein, it is to be understood that the present disclosure is intended to teach, suggest, and illustrate various features and aspects of the invention, but is not intended to be limiting with respect to the scope of the invention, as defined exclusively in and by the claims, which follow.

Indeed, it is contemplated and to be explicitly understood that various substitutions, alterations, and/or modifications, including but not limited to any such implementation variants and options as may have been specifically noted or suggested herein, including inclusion of technological enhancements to any particular method step or system component discovered or developed subsequent to the date of this disclosure, may be made to the disclosed embodiment of the invention without necessarily departing from the technical and legal scope of the invention as defined in the following claims. 

1. A football adapted to be thrown in a manner resulting in spinning of said football substantially around a long axis thereof as said football travels along a trajectory; said football having spin speed indicia applied on an outer surface, said indicia being aligned with the direction of spin of said football about said long axis; wherein said indicia provide an indication of whether said football is spinning at at least a first predetermined minimum spin speed.
 2. The football of claim 1, wherein said indicia comprise a first circumferential indicator ring on said outer surface.
 3. The football of claim 2, wherein said first circumferential indicator ring comprises a plurality of alternatingly-colored segments of a first color and a second color, said segments forming a mixed color when said football spins at at least said first predetermined minimum speed.
 4. The football of claim 3, wherein the width of said segments is selected to establish a said first predetermined minimum spin speed.
 5. The football of claim 31, wherein said indicia further comprise a second circumferential indicator ring on said outer surface.
 6. The football of claim 5, wherein said second indicator ring provides a single reference color relative to said mixed color of said first indicator ring.
 7. The football of claim 6, wherein said second indicator ring comprises a plurality of alternatingly-colored segments.
 8. The football of claim 6, wherein said second indicator ring comprises a plurality of alternatingly-colored segments of said first color and said second color.
 9. The football of claim 7, wherein said alternatingly-colored segments of said second indicator ring have a different width then said alternatingly-colored segments of said first indicator ring.
 10. The football of claim 7, wherein said second indicator ring appears to be a single, mixed color when said football spins at at least a second predetermined minimum spin speed, said second predetermined minimum spin speed being less than said first predetermined minimum spin speed.
 11. A method of providing a visual indication of the spin speed of a football adapted to be thrown in a manner resulting in spinning of said football substantially around a long axis thereof as said football travels along a trajectory, the method comprising: applying spin speed indicia on an outer surface of said football, said indicia being aligned with the direction of spin of said football about said long axis. wherein said indicia provide an indication of whether said football is spinning at at least a first predetermined minimum spin speed.
 12. The method of claim 11, wherein said indicia comprise a first circumferential indicator ring on said outer surface.
 13. The method of claim 12, wherein said first circumferential indicator ring comprises a plurality of alternatingly-colored segments of a first color and a second color, said segments forming a mixed color when said football spins at at least said first predetermined minimum speed.
 14. The method of claim 13, wherein the width of said segments is selected to establish a said first predetermined minimum spin speed.
 15. The method of claim 13, further comprising applying a second circumferential indicator ring on said outer surface.
 16. The method of claim 15, wherein said second indicator ring provides a single reference color relative to said mixed color of said first indicator ring.
 17. The method of claim 16, wherein said second indicator ring comprises a plurality of alternatingly-colored segments.
 18. The method of claim 16, wherein said second indicator ring comprises a plurality of alternatingly-colored segments of said first color and said second color.
 19. The method of claim 17, wherein said alternatingly-colored segments of said second indicator ring have a different width then said alternatingly-colored segments of said first indicator ring.
 20. The method of claim 7, wherein said second indicator ring appears to be a single, mixed color when said football spins at at least a second predetermined minimum spin speed, said second predetermined minimum spin speed being less than said first predetermined minimum spin speed. 